The graded Chern character respects relative maps and skeletal filtrations

Use $i:A_{+}\to X_{+}$ and its reduced mapping cone $C_i=X_{+}{\cup }_{i}C{A}_{+}$. All these are finite based CW complexes with vertex basepoints. The contractible cone is a CW subcomplex of $C_i$. Its contraction to the cone tip extends over $C_i$ by [F4]; at the final time the extension is constant on the cone and factors through its collapse $r:C_i\to Q$. The extended homotopy, and its quotient homotopy, show that $r$ is a based homotopy equivalence. For $A=\mathrm{\varnothing }$, $A_{+}=\ast$, the reduced cone is a point and $C_i=Q=X_{+}$. Thus [F2] identifies $K^j(X,A)$ with ${\tilde{K}}^j(C_i)$ in all degrees, using the absolute convention when $A$ is empty.

Let $\alpha \in F^p{K}^j(X)$. Naturality for the inclusion $X^{p-1}\hookrightarrow X$ gives ${\mathrm{ch}}^j(\alpha ){\mathrm{\mid }}_{X^{p-1}}={\mathrm{ch}}^j(\alpha {\mathrm{\mid }}_{X^{p-1}})={\mathrm{ch}}^j(0)=0$. By the displayed kernel definition this is exactly ${\mathrm{ch}}^j(\alpha )\in F^{p}H{P}^j(X;\mathbb{Q})$. Only the actual K-theory and ordinary restriction maps are used.

The analogous ordinary identification is $H^m(X,A;\mathbb{Q})\cong {\tilde{H}}^m(C_i;\mathbb{Q})$. For nonempty $A$, view $C_i$ as $X$ with the unreduced cone on $A$ attached, based at its tip. Excision identifies $H^m(C_i,CA)$ with $H^m(X,A)$: remove the closed upper part of the cone at heights at least $1\mathrm{/}2$, which lies in the interior of $CA$; the remaining pair is $X$ with a collar $A\times [0,1\mathrm{/}2)$ and deforms as a pair to $(X,A)$. Since $CA$ is contractible, the pair sequence identifies $H^m(C_i,CA)$ with the kernel of restriction from $H^m(C_i)$ to the tip, including degree zero. For empty $A$, finite disjoint additivity identifies ${\tilde{H}}^m(X_{+})=H^m(X)$, also when $X$ is empty. These identifications and $r^{\ast }$ are natural because the maps of cones and the quotient pullbacks are natural; inverses of natural isomorphisms are natural.

Write $q:C_i\to \mathrm{\Sigma }{A}_{+}$ for collapse of $X_{+}$. Normalize the pair connector in K-theory as ${\mathrm{\partial }}_K=-q^{\ast }{s}_K$ under step 1.1. This differs by a sign from the cofiber arrow $q^{\ast }$ and is still natural and exact by [F2]. The sign matches the ordinary pair connector. Indeed the ordinary suspension $s$ is the boundary for $(C{A}_{+},A_{+})$, with the tip at height one. Extend a cocycle $a$ on $A$ to a cochain $b$ on $X$, and extend it to $c$ on the cone with value zero at the tip. The ordinary pair boundary is represented by $db$ on $(X,A)$, while $q^{\ast }s(a)$ is represented by $dc$ on the cone and zero on $X$. Their sum is the coboundary of the glued cochain $(b,c)$ on the cone attachment; therefore their reduced classes are negatives. This computation can be made on cochains subordinate to the cone-collar cover: subdivision and excision from [F3] identify them with singular cohomology, and restrictions on the overlap are precisely the common cochain $a$. The quotient for the cone class is $C{A}_{+}\mathrm{/}{A}_{+}=\mathrm{\Sigma }{A}_{+}$, including the tip in the collapsed basepoint. Consequently ${\mathrm{\partial }}_{HP}=-q^{\ast }{s}_{HP}$. For empty $A$ the sources are zero. The computation is degreewise; summing over even shifts gives the periodic pair sequence. Only finitely many degrees contribute on each finite CW pair.

Define the relative character by the based character on $C_i$, transported through steps 1.1 and 2.1. It is natural for maps of pairs by [F1] and the natural cone maps. Naturality with respect to $q$ and suspension compatibility give ${\stackrel{}{\mathrm{ch}}}^{j+1}{q}_K^{\ast }{s}_K=q_{HP}^{\ast }{\stackrel{}{\mathrm{ch}}}^{j+1}{s}_K=q_{HP}^{\ast }{s}_{HP}{\stackrel{}{\mathrm{ch}}}^j$. Negating this equality and using step 3.1 gives the asserted connector identity. The other two maps in the pair sequence are induced by the inclusion and quotient maps, so commute by the same naturality. When $A=\mathrm{\varnothing }$, the construction is exactly the absolute character on $X_{+}$; no based structure on empty $X$ is assumed.

For $A=X$ the cofiber is contractible and both relative groups are zero. For $X=\mathrm{\varnothing }$, both absolute groups are zero by their disjoint-basepoint conventions. If $p\le 0$, the filtration is the whole group because the target is the group of the empty space; if $p>\dim X$, it is zero because restriction is the identity. Negative and positive $j$ use the same Bott-compatible suspension construction in [F1]–[F2], so the identities persist in every degree. This proves all claims.